Problem source

A needle of length two cm is dropped at random onto a large piece of paper ruled with parallel lines two cm apart.
\begin{questionparts}
	\item By considering the angle which the needle makes with the lines, find the probability that the needle crosses the nearest line given that its centre is $x$ cm from it, where $0 < x < 1$.
\item Given that the centre of the needle is $x$ cm from the nearest line and that the needle crosses that line, find the cumulative distribution function for the length of the shorter segment of the needle cut off by the line.
\item Find the probability that the needle misses all the lines.
\end{questionparts}

Solution source

\begin{questionparts}
\item Suppose the needle's center is $x$ cm from the nearest line and makes an angle of $\theta$. Then if $\sin \theta > x$ it will cross the line, otherwise it will not.

Given that $\theta \sim U(0, \frac{\pi}{2})$, we can see that 

\begin{align*}
&& \mathbb{P}(\text{needle crosses}) &= \mathbb{P}(\sin \theta > x) \\
&&&= \mathbb{P}(\theta > \sin^{-1} x) \\
&&&= 1-\frac{2\sin^{-1} x}{\pi}
\end{align*}

\item The length of the short segment is $L = 1 - \frac{x}{\sin \theta}$ and $\theta \sim U(\sin^{-1} x, \frac{\pi}{2})$.

So \begin{align*}
&& F_L(l) &= \mathbb{P}(L < l) \\
&&&= \mathbb{P}\left (1 - \frac{x} {\sin \theta} < l\right) \\
&&&= \mathbb{P}\left ( \sin \theta < \frac{x}{1-l}\right) \\
&&&= \mathbb{P}\left (\theta  < \sin^{-1} \frac{x}{1-l}\right) \\
&&&= \frac{ \sin^{-1} \frac{x}{1-l} - \sin^{-1} x }{\frac{\pi}{2} - \sin^{-1}x}
\end{align*}

\item The needle (with probability $1$) cannot hit $2$ lines, so let's only consider the line it's nearest too. The distance to this line is uniform on $[0,1]$, and the so we want to calculate.

\begin{align*}
&& \mathbb{P}(\text{needle crosses}) &= \int_0^1 \left (1 - \frac{2\sin^{-1}x}{\pi} \right) \d x \\
&&&= 1 - \frac{2}{\pi} \int_0^1 \sin^{-1} x \d x\\
&&&= 1 - \frac{2}{\pi} \left ( \frac{\pi}{2} - 1 \right) \\
&&&= \frac{2}{\pi} 
\end{align*}

Therefore the probability it misses is $1 - \frac{\pi}{2}$

\end{questionparts}